1. Introduction

In the past decades, the optical scan solutions with compact size, low power, large field-of-view (FOV) and high scan speed have attracted many interests, such as MEMS mirror, scanning fiber and optical phased array [1–3], which have been developed to realize the wide-angle beam pointing or imaging adjustment. But the scan precision of these scan techniques mostly depends on the accuracy of the driver, for example, the resolution of MEMS actuator or piezo tube actuator. In fact, in order to effectively steer the laser beam to track and point a dynamic object with high accuracy, some scanning systems with Risley prisms have also become promising solutions. Risley prism laser-beam steering systems can provide a robust alternative for portable and mobile systems. Compared with other optical scanning technologies, the scan precision of the Risley-prism scanner can be higher than the driver accuracy owing to the prism refraction principle. Besides, the FOV and the scan speed of the Risley prisms can be up to 120° and better than 3000 rpm, respectively [4–6]. Therefore, Risley-prism systems are widely employed in the fields of laser communications, space observation, infrared countermeasure, military weapons, biomedicine, machining, searching and rescuing [4–8].

Conceptually, the laser beam steering system is composed of two identical Risley prisms, and the incident beam can be steered to any point within a cone by adjusting the rotation angles of two prisms [9]. With different speed combinations of two prisms, various beam scan modes can be provided throughout the full field. However, Risley prisms still have to face several problems. Some are associated with optical design, such as wavefront quality, beam compression, and chromatic dispersion, which have been studied in many publications [10–13]. In most of these works, the approximate solutions of the beam steering system are especially concerned based on two crucial factors of wedge angle and refractive index of two prisms, while the influence of system structural parameters on the exact solutions is scarcely ever considered. Actually, three important issues, i.e., the blind zone, the nonlinear problem and the scan precision, are inevitable to be confronted for the design of Risley-prism system. Considering structural parameters, a blind zone will occur at the center of the scan region, and the nonlinear problem of beam deviation can generate a singularity, both of which can shrink the scan region and even lead to the target loss. Moreover, a high-accuracy scan theory of Risley-prism scanner should be verified, and then a better scan performance can be validated through the effective nonlinear motion control method.

Although some previous literatures have described ways to achieve a fully-covered scan region [5,14], the scan blind zone resulting from structural parameters are not further explored. Without consideration of the structural parameters, the beam exiting point on the second prism is regarded as the center point of the emergent surface and then an approximate solution can be obtained without a blind zone [5,14]. However, the structural parameters should not be ignored in some applications of near distance scanning. An inherent blind zone resulting from the structural parameters exists in Risley-prism systems, which to some extent hinders the beam experience in full field [15,16]. For an available guidance on optical design, a quantitative analysis method should be established to investigate the impact of system parameters on the blind zone size. And another essential issue is to reveal the nonlinear change mechanism of the emergent beam deviation angle, so as to formulate some effective control strategies for the beam steering. Efforts aforementioned have provided effective and efficient analytical solutions to the slew rate of the emergent beam and the required angular velocities of two prisms [17], but in order to effectively measure the dynamic targets, the relation between the moving rate of the pointing target and the angular velocities of prisms should be further studied. Besides, most previous works concentrate on the relation between the scan patterns and the prisms rotation control [18,19], while the beam scan precision, as a key index for a beam scanning system, rarely catches the eyes of researchers.

The organization of the paper is as follows: Section 2 expounds the theoretical derivation of the beam scan model and the scan blind zone for a Risley prism system. Section 3 reveals the nonlinear scan mechanism between the prisms’ speeds and the pointing target velocity, and especially explores the singularity issue during the beam tracking process. The scan precision of beam deviation along the radial and circumferential directions are compared to each other and the high-accuracy beam steering mechanism is verified in Section 4. Conclusions are drawn in Section 5.

2. Beam scan patterns and scan blind zone in rotation double-prism system

2.1 Modelling of rotation double-prism system

Under Cartesian coordinate system XYZ established in Fig. 1, rotation double-prism system consists of two identical prisms, each of which is designated with wedge angle α, refraction index n, thinnest-end thickness d₀ and clear aperture D_p. As one of the four configurations proposed by Li [20], the plane surfaces of two prisms are situated outward. Two prisms, sequentially named as prism 1 and prism 2 along the positive direction of Z axis, can rotate about the Z axis at arbitrary angular velocities ω₁ and ω₂, respectively, where the clockwise rotation angle is defined as the positive angle [18]. The prism surfaces and the screen are labeled as Σ₁₁, Σ₁₂, Σ₂₁, Σ₂₂ and P, the central point of which are marked as O₁₁, O₁₂, O₂₁, O₂₂ and O_P in sequence.

 

Fig. 1 Schematic diagram of rotation Risley-prism system.

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On the initial state, the principal section of each prism locates in the XOZ plane with the thinnest end towards the positive direction of X axis, and the rotation angles denoted by θ₁ for prism 1 and θ₂ for prism 2 are both equal to 0°, which act as the variables θ₁(t) and θ₂(t) with time change. For convenience, other notations are also involved as follows. D₁ is the distance between two prisms, and D₂ is that from prism 2 to the screen P. Supposed that the laser beam propagates along the positive direction of Z axis, it passes through the intersection points O₁, O₂, K and N on surfaces Σ₁₁, Σ₁₂, Σ₂₁ and Σ₂₂ sequentially, and finally arrives at the scan point M_P on the screen P. The point N_p is the projection of the beam point N on screen P. The pitch angle ρ represents the deviation angle of the emergent beam vector with respect to the positive direction of Z axis, and further, the azimuth angle φ represents the separated angle between the emergent beam vector projected on the screen P and the positive direction of X axis.

According to the vector refraction theorem, the pitch angle ρ can be expressed as follows [11],

The pitch angle ρ of the emergent beam reaches the maximum when the separated angle between two prisms is zero, and decreases to zero when the separated angle becomes 180°. Thus, the beam can be steered to any point within a cone with a half-angle equal to the maximal pitch angle by adjusting the rotation angles of two prisms. Through simulation with different constant ratio ω₁/ω₂, the corresponding scan trajectory on the screen can be variously produced. If ω₁/ω₂ exceeds zero, the scan trajectory resembles a large circle with some smaller inscribed ones. If ω₁/ω₂ is otherwise less than or equal to zero, a petal-like trajectory is generated. Particularly, the trajectory is a standard circle when ω₁/ω₂ = 1 or an ellipse similar to a straight line when ω₁/ω₂ = −1. These are summarized as the basis to study the scan trajectories of Risley-prism system [18,19].

2.2 Blind zone in scan region

Determined by the structural parameters of rotation double-prism system, the scan point M_p (x_p, y_p, z_p) of the emergent beam never covers the coordinate origin on the screen, which illustrates a fact that an existing blind zone, located at the center of scan region, probably causes the loss of target during searching and tracking applications. In Fig. 2, a blind zone is viewed in a rotation double-prism system with α = 10°, n = 1.517, d₀ = 10 mm, D_p = 400 mm, D₁ = 400 mm and D₂ = 400 mm. In order to keep the emergent beam inside the clear aperture during the beam propagation, the separated distance of two prisms D₁ ranges between 161 mm and 2187 mm.

 

Fig. 2 The scan range of the rotation double-prism system, where a blind zone locates in the central region

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The distance |M_PO_P|, as indicated in Fig. 3, is correlated to the separated angle |∆θ| between two prisms, and the minimum of |M_PO_P| is defined as the radius R of blind zone, which can be obtained under two conditions, i.e., the pitch angle ρ = 0 and the distance |NO₂₂| reaches its minimum. Besides, given a specified Risley-prism system, the structural parameter D₁ can determine the distance |NO₂₂|, while another parameter D₂ can be considered as the weight of ρ. In order to investigate the forming law of the blind zone radius R, a quantitative method is proposed as follows.

 

Fig. 3 The relation between |M_pO_p| and |∆θ| when (a) D₁ = 400 mm and (b) D₁ = 2100 mm. (c) and (d) are the enlarged view of area M and N, respectively.

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As an example, the above rotation double-prism system is investigated. Upon employing one-dimensional search method throughout the accessible range of D₁ (i.e., 161 to 2187 mm), a unique minimum of |M_PO_P| can be determined when D₁ varies from 161 to 1181 mm, accompanied by |∆θ| = 180°. In Fig. 3(a) the minimal |M_PO_P| of 29.036 mm occurs when D₁ = 400 mm and |∆θ| = 180°, and the pitch angle of the emergent beam equals to zero. Moreover, as shown in Fig. 4, when D₂ changes, the minimal |M_PO_P| remains unchanged.

 

Fig. 4 The relation between R and D₂ when (a) D₁ = 400 mm and (b) D₁ = 2100 mm.

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Nevertheless, when D₁∈[1181,2187] mm, the minimal |M_PO_P| is related to the factor D₂. For instance, in the case of D₁ = 2100 mm and D₂ = 10 mm, as shown in Fig. 3(b), the minimal |M_PO_P| is 188.295 mm where |∆θ| = 130.9° or 229.1°. But in the case of D₁ = 2100 mm and D₂ = 400 mm, the unique minimal |M_PO_P| is 188.54 mm where |∆θ| = 180°. Moreover, to present the correlation of D₂ to the minimal |M_PO_P| (namely, the blind zone radius) in Fig. 4(b), D₁ is thus set to 2100 mm as a constant. It is illustrated that with the growth of D₂ within the range of (1, 25) mm, the minimal |M_PO_P| occurs on two sides of |∆θ| = 180° and increases from 187.92 to 188.54 mm. In contrast, when D₂≥25 mm, the minimal |M_PO_P| equals to 188.54 mm where |∆θ| = 180°, and the blind zone radius has no changes.

Obviously, when D₁∈[161, 1181] mm, the distance |NO₂₂| can reach the minimum and meanwhile the pitch angle ρ equals 0. As a result, D₂ has nothing to do with the radius R of the blind zone. However, when D₁∈(1181, 2187] mm, two aforementioned conditions (ρ = 0 and minimum |NO₂₂|) cannot be reached simultaneously. The impact of ρ on radius R is relative to D₂. With the increase of D₂, the impact of ρ will be beyond that of D₁ on R. Hence, the minimal |M_PO_P| can be obtained only when ρ = 0.

In general, given the wedge angle α, the refractive index n, and the thinnest-end thickness d₀, according to the distance threshold between two prisms and that of prism 2 separated from the screen, written as D_1c and D_2c, respectively, the change rules of the blind zone radius can be summarized as follows.

3. Nonlinear problems of beam scanning

3.1 Nonlinearity analysis

In fact, the function relation is nonlinear between the prism motion and the deviation of the emergent beam, which can be summarized as two issues: the rotation angles of prisms relative to the beam deflection angle, and the rotation speeds of prisms relative to the beam slew rate [21].

In order to investigate the nonlinearity, the slew rate of the emergent beam is resolved into the tangential slew rate ω_t and the radial slew rate ω_r [17]. They can be expressed as the time derivative of azimuth angle φ and pitch angle ρ, respectively, as follows.

According to the second step of the two-step method [11,17], when the emergent beam reaches the expected pitch angle ρ, the azimuth angle φ can be achieved by simultaneously rotating two prisms with a constant separated angle ∆θ. That is to say, the ratios from the angular velocities of prisms to the change rate of azimuth angle φ keep uniform and are equal to 1. Therefore, we just consider the relation between the angular velocities of prisms and the change rate of pitch angle ρ, as indicated in Fig. 5. Here the wedge angle α is set to 5°, 10° or 15° in turn, the pitch angle ρ varies from 0° to the maximum ρ_max, while the azimuth angle φ remains invariable.

 

Fig. 5 The nonlinear relation between the angular velocities of prisms and the change rate of pitch angle dθ/dρ.

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In Fig. 5, the angular velocities of prisms change greatly with the pitch angle ρ ranging from 0° to ρ_max. When the pitch angle ρ is close to ρ_max, the ratio of the angular velocities to the pitch angle even tends to infinity. Thus in order to actualize the cooperative motion of target and prisms, the velocities of prisms are required to be strictly regulated all the time, which poses challenges for the motion control of the whole system.

3.2 Singularity analysis

The angular velocities of prisms must increase dramatically whenever the beam scan trajectory approaches the inner or outer edge of the scan region, so two driving motors should possess the ability of strong accelerations. In other words, the smooth and steady beam steering along a continuous trajectory is hardly executed in the center and outer edge of the scan region, which is called the singularity problem of rotation Risley-prism system [17].

Based on Eq. (7), the tangential slew rate ω_t and radial slew rate ω_r is further provided as

Hence, the relations of ω_t and ω_r relative to the moving velocities of the scan point along the X and Y axis are individually given by

As shown in Figs. 6 and 7, when the beam scan trajectory approaches the inner edge of the scan region, ω_t is nearly infinite while ω_r is close to zero. At this time, the angular velocities of prisms overwhelmingly contribute to the tangential slew rate ω_t rather than ω_r. In other words, the angular velocities are required to be infinite in theory, which results in a singularity problem at the center of the scan region. In fact, for other beam scan schemes such as raster scanning and Lissajous scanning [22,23], a singularity problem is ubiquitous at the center of the scan region, where a high tangential slew rate is necessary.

 

Fig. 6 The relations between the tangential slew rate ω_t of emergent beam and the scan point moving velocity v_x and v_y along X axis and Y axis, respectively. (a) ω_t/v_x; (b) ω_t/v_y.

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Fig. 7 The relations between the radial slew rate ω_r of emergent beam and the scan point moving velocity v_x and v_y along X axis and Y axis, respectively. (a) ω_r/v_x; (b) ω_r/v_y.

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However, for Risley-prism scanner, singularity problems occur not only at the inner edge, but also the outer edge of the scan region. As shown in Fig. 5, when the beam scan trajectory approaches the outer edge of the scan region, the ratios of angular velocities of prisms to ω_r tend to infinity while the ratios of angular velocities of prisms to ω_t are 1 according to the two-step method. Meanwhile, the tangential slew rate ω_t and the radial slew rate ω_r draw near to fixed values. Therefore, the angular velocities of prisms mostly contribute to the radial slew rate ω_r rather than ω_t, and are theoretically infinite.

With a specified maximum speed of the target, infinite angular velocities of prisms are needed for the Risley-prism scanner to catch the target when it passes through the inner and the outer edge of the scan region. Therefore, the smooth and steady tracking is difficult to implement at these two areas due to the limited maximum speeds of driving motors, which can lead to the target loss. Actually, the size of these two areas mainly depends on the distance D₂ between prism 2 and the receiving screen. Given an active scope of the target, an appropriate distance D₂ should be selected to avoid the singularity problem.

4. Scan precision analysis for emergent beam

In rotation Risley-prism systems, the position of a beam scan point is defined by the pitch angle ρ of the emergent beam that describes its radial position and the azimuth angle φ that describes its circumferential position.

The emergent beam vector deduced from Eq. (1) is given by

And the azimuth angle φ is written as

Hereby, the scan precision is investigated along the circumferential and radial direction, respectively.

4.1 Circumferential precision analysis of emergent beam

To evaluate the circumferential scan precision for rotation Risley-prism system, the change rate of the azimuth angle φ is investigated when the system produce a circular trajectory on the screen during one revolution. Here the separated angle Δθ between two prisms keeps invariable such that the azimuth angle φ of emergent beam depends only on the rotation angle of prism 2.

Providing Δθ is the unique influential factor on the deflection angle δ₁ by prism 1 as well as δ₂ by prism 2, we can find that A = sinδ₁cosδ₂cosΔθ_r + cosδ₁sinδ₂ and B = sinδ₁sinΔθ are both constants. Thus, with components of the emergent beam written as x_f = −Acosθ₂ + Bsinθ₂ and y_f = −Asinθ₂−Bcosθ₂, respectively, the equation below is satisfied.

Since the azimuth angle φ as the function of the rotation angle θ₂ is expressed by φ = f(θ₂), the azimuth angle error δ_φ can be calculated from

If y_f ≥ 0, the change rate dφ/dθ₂ of the azimuth angle is derived as

And when y_f < 0, the case can be written as

It is worth mentioning that the change rate of the azimuth angle of the emergent beam equals to 1 because of the constant separated angle between two prisms, which indicates an equal change in the azimuth angle φ and the rotation angles θ₁ and θ₂. Namely, in rotation double-prism systems, the circumferential scan precision always keeps consistent with the precisions of rotation angles of prisms.

4.2 Radial precision analysis of emergent beam

For the radial scan precision of the emergent beam, the pitch angle error δ_ρ is determined by

The wedge angle error δ_α and refraction index error δ_n are both classified as systematic errors, except for the random error in the separated angle between two prisms δ_Δθ. In some specific Risley-prism systems, δ_Δθ occurs as a single factor that affects the pitch angle error δ_ρ of the emergent beam.

In order to clarify each impact of δ_α, δ_n and δ_Δθ on δ_ρ, the partial derivatives of pitch angle ρ expressed by$\left|\frac{\partial \rho }{\partial \alpha }\right|$, $\left|\frac{\partial \rho }{\partial n}\right|$ and $\left|\frac{\partial \rho }{\partial \Delta \theta }\right|$are individually plotted in Fig. 8, where the separated angle ∆θ between two prisms acts as an independent variable. In Figs. 8(a) and 8(b), the correlation curves of $\left|\frac{\partial \rho }{\partial \alpha }\right|$ and $\left|\frac{\partial \rho }{\partial n}\right|$ with respect to ∆θ display some essential similarities, such as the symmetric distributions about ∆θ = 0° and the gradual increases with ∆θ varying from −180° to 0°. Meanwhile, larger wedge angle α or larger refraction index n will apparently increases δ_ρ corresponding to δ_α or δ_n, respectively. It is somewhat different in Fig. 8(c) that the correlation curves of $\left|\frac{\partial \rho }{\partial \Delta \theta }\right|$ relative to ∆θ are also symmetric about ∆θ = 0°, but decrease monotonically as ∆θ varies from −180° to 0°. The δ_ρ corresponding to δ_∆θ increases as a result of the increasing n or α, anyway. The larger n or α is, the worse the radial scan precision becomes, which in contrast contributes to a larger scan region. Thus a balance must be made between beam scan range and scan precision during the selection of the material and the wedge angle α.

 

Fig. 8 The effects of a prism parameter ∆θ on the partial derivatives of pitch angle, including $(a)\left|\frac{\partial \rho }{\partial \alpha }\right|,(b)\left|\frac{\partial \rho }{\partial n}\right|and(c)\left|\frac{\partial \rho }{\partial \Delta \theta }\right|.$

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Specifically, if α = 10°, n = 1.517 and Δθ ranges from −180° to 180°, the maximum values of $\left|\frac{\partial \rho }{\partial \alpha }\right|$ and $\left|\frac{\partial \rho }{\partial n}\right|$are 1.077 and 0.358 when Δθ = 0°, but the maximum value of $\left|\frac{\partial \rho }{\partial \Delta \theta }\right|$ equals to 0.092 when Δθ = 180°. Under the assumptions that the manufacturing error of wedge angle δ_α approximates 1″, the refraction index error δ_n resulting from inhomogeneous optical glasses is up to ± 1 × 10⁻⁵ and the relative rotation angle error of prisms δ_Δθ achieves 0.01°, it can be calculated that the maximum of δ_ρ caused by each single factor among δ_α, δ_n and δ_Δθ reaches about 5.22 μrad, 3.58 μrad and 16.06 μrad in sequence. Consequently, the high precision for target tracking can be guaranteed in principle owing to the large reduction ratio from the separated angle ∆θ between prisms to the pitch angle ρ of the emergent beam.

5. Conclusions

Rotation double Risley-prism scanner is one of multi-mode scan methods. The paper thoroughly presents some important issues for the exact beam scanning in double Risley-prism scanning system, including scan blind zone, nonlinear scan, singularity and scan precision. The beam steering performance determined by the physical parameters is described by the quantitative analysis method. Considering the nonlinear beam scanning of the emergent beam, a nonlinear control strategy must be built in order to steer the beam as required for synchronously tracking the moving target. Due to the scan singularity, a smooth and steady tracking process cannot be implemented when the emergent beam moves through a specific range at the inner or outer edge of scan region, where the actual scan region shrinks compared with the theoretical one. Owing to a large reduction ratio from the rotation angles of prisms to the radial deflection angle of the emergent beam, the high-accuracy radial beam steering can be obtained.

In fact, the blind zone and singularity problem will both limit the scan region, which can be mitigated by adding a third prism into the existing Risley-prism system [16,24]. In the future, a triple Risley-prism system with a nonlinear control strategy will be developed by us to improve the performance of Risley-prism scanner, and the design principle based on the parameter selection proposed in this paper can be referred for further study.